Bergman Geodesics

Review article, 2012

The aim of this survey is to review the results of Phong-Sturm and Berndtsson on the convergence of Bergman geodesics towards geodesic segments in the space of positively curved metrics on an ample line bundle. As previously shown by Mabuchi, Semmes and Donaldson the latter geodesics may be described as solutions to the Dirichlet problem for a homogeneous complex Monge-Ampere equation. We emphasize in particular the relation between the convergence of the Bergman geodesics and semi-classical asymptotics for Berezin-Toeplitz quantization. Some extension to Wess-Zumino-Witten type equations are also briefly discussed.